17 MEKOMORPHIC FUNCTIONS 249

£//(f)\ 1g(-l)
dz \sin 27 ^ (z —

n-nf

where the series on the right-hand side is normally convergent in any compact subset
of C which does not contain any of the points mr (n integer). (Consider the integral

2m J sin x (x — z)2

Vn

where yn is the circuit /->(« + %)Trelt for —Tr^t^TT. Observe that for every e > 0,
there is a number c(e) > 0 such that the relations \z — mr\ ^ e for every integer n e Z
imply |sin z 5* c(e)el>J<f<z)l; and use the theorem of residues.)
3. (a) Show that for Z^HTT (n integer)

1 * 2z
cotz--+£-----—

Z n= 1 Z2 — «27T2

where the right-hand side is normally convergent in any compact subset of
C which does not contain any of the points mr. (Use Problem 2 and the relation
lim (cotz-l/z) = 0.)

(b) Deduce from (a) that

00 ( z2 \

~ \ I

n / z2 \

where the product is defined as the limit of Yl 11 "~ TT~I I» tne convergence being

*-i\ k TT /•

uniform in every compact subset of C. (Consider the entire function

+ 00 / z\
f(z)=V[ 1------)e-*/n* (Section 9.12, Problem 1), and use (a) to prove that the

-oo \ mr]

function (sin z)/z/(z) is a constant.)
(c) Deduce from (b) the identity

1 1 .

—- = — sm -TTZ

(see Section 9.12, Problem 2).

4. Let/be a complex valued function analytic in an open neighborhood A of 0 in Cp;
for convenience we write w instead of zp and z instead of (zt,..., zp_0. Suppose that
/(0,0) = 0 and that the function /(O, H>), which is analytic in a neighborhood of H> — 0
in C, is not identically 0. Then there exist an integer r > 0, r functions /z/z) analytic in
a neighborhood of 0 in C77"1, and a function ^(z, w) analytic in a neighborhood B <= A,
of 0 in Cp and ^ 0 in that neighborhood, such that

/(z, w) = (>/ + hi(z)Yf~l + • • • H- hr(z))g(z, w)

in a neighborhood of 0 in Cp (the "Weierstrass preparation theorem"). (If /(O, w)
has a zero of order r at w = 0, use (9.17.4) to prove that there is a number e > 0 and
a neighborhood V of 0 in Cp~ * such that for any z e V, the function w> ->/(z, w)
has exactly r zeros in the disc |w| < e and no zero on the circle |w| = fi. Let y be the
circuit t-*eelt for — TT^ / ^ TT; using the theorem of residues, show that there ares not a pole